Lets talk about the base ten number system a little. It's the number system we normally use and has 10 digits 0,1,2,3,4,5,6,7,8,9. That is why we call it the base ten number system. There are other number systems you will use later on too. Binary or base 2 only having 2 digits, 0 and 1. Octal only having eight digits 0,1,2,3,4,5,6,7. Hexadecimal having 16 digits 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F.
In our regular everyday base ten system, we can count to 9 easy enough having a digit to represent each value. When we want to count higher than 9 and only can use the ten existing digits, we start using place values and the same set of digits over and over. One more than 9 is 10. 0 Is in the one's place value and tells us how many ones we have. 1 is in the ten's place value and tells us how many times we have 1 full set of 10 one values. The illustration below shows counting from 10 to 20 and the place values.
If we continue counting past 20 and up to 30, we just add 1 to the one's place each count until we reach 9. When we reach 9 in the one's place and add one, we change the one's place to 0 and add 1 to the ten's place. Showing we have 3 sets of 10 values and no or 0 one values. The expanded form of a number works like the illustration above only we use numbers and operators(+,-,X,/) to show the numbers and place values. For simplicity look at the ten's and one's place as we count from 0 to 20 showing the expanded form.
If we continue counting until we reach a value of 99 we would add 1 to 9 in the one's place value then change the one's to 0 and add 1 to the ten's. Since 9 in the ten's place value is the highest digit we can use, we change it back to 0 and add 1 to the hundred's place value giving us 100. The expanded form of 100 is (100 x 1) + (10 x 0) + (1 x 0). As we continue counting we find that each place value to the left of the one's place value can be found by multiplying 10 time itself the number of the positions to the left of the one's place. For example 1 in the number 10 is in the ten's position and is one place to the left of the one's place so 10 x 1 = 10. In the number 100(one hundred) 1 is 2 positions left of the one's place so its value is (10 x 10). In the number 1000, 1 is 3 positions to the left of the one's place so its value is (10 x 10 x 10) or one thousand. This pattern just keeps on going to infinity as high as you may want to count. When we add numbers with multiple digits as in these exercises we write the numbers down one on top of the other making sure to line up the one's place on top of the one's place in all the numbers to be added so all the place values of each number in the stack line up with each other. Then just as in counting we start at the far right or one's place in these exercises and add up the numbers in that column. If you reach a number greater than 9 as you add up the numbers in the one's place column, place the one's part of that number under the one's column and carry the higher place values to the ten's place column and add the value to the ten's column of the numbers you are adding.
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Screenshots using Homemade Math's built in scratchpad tiles to solve multiple digit addition problems. |
Above 3 is added to 9 resulting in the sum of 12. The 2 is written directly under the one's place column of the numbers being added and the 1 is carried and placed on top of the ten's place column of the numbers being added and included in the sum of the ten's place column . This 1 that is carried and added represents a value of 10 and is the same as adding 10 to the number because of it is one place left of the one's place in the numbers being added.
Now we add the carry of 1 plus the numbers in the ten's column and put the sum of 7 directly under the ten's column. This time we don't have a carry to worry about. When we add the 1,1 and 5 in the ten's column we are really adding 10, 10 and 50 because of there place in the ten's column.
Lastly we add 4 and 3 for a sum of 7 and place a 7 directly under the hundred's place of the numbers being added. Because the 4 and 3 are in the hundred's place we are really adding 400 and 300. Here are a few more examples.
Adding 8 and 3 in our one's place gives us 11 so we place a 1 directly under the one's column and carry a 1 to the ten's column, placing it over the 5 and 5. Add 1, 5 and 5 in our ten's column, we arrive at 11 again. Place a 1 directly under the ten's column and carry a 1 to the hundred's column placing it over the 7. Since we are adding in the ten's column, we are really adding 10, 50 and 50. The two 50s added together make a whole 100 so we put a 1 over the 7 as our carry because the 7 is really 700 and put a 1 under the two 5s to show our ten's. Last we add our 100 column up adding 1 and 7 for a sum of 8. We place the 8 directly under the hundred's column. Our final answer is 811. Eight hundred and eleven.
First add the one's column getting 6 as a result. Place the 6 under the 4 and 2 in the one's column. There is no carry. Add the ten's column 9 and 2 getting 11 as a result. Place a 1 under the ten's column and carry a 1 placing it over the one hundred's column. Add the hundred's column 1, 9 and 9 for a result of 19. Place the 9 under the hundred's column and carry the 1 to the thousand's column. There are no numbers in the thousand's column so we just bring our 1 down and place it in front of the 9 for a final sum of 1916 One thousand nine hundred and sixteen.
Add 6 and 9 in the one's column for a result of 15. Place the one's part of 15 under the one's column and carry the ten's part over to be added the ten's column. Add 1(the carry), 8 and 4 in the ten's column resulting in 13. Place the one's part of 13 which is 3 under the ten's column and carry the ten's part of 13 which is 1 over to the hundred's column. Since we are adding the ten's column the 3 is really a value of thirty(30) meaning there are 3 full sets of ten. The carry of 1 over to the hundred's column has a value of 100(one hundred). Next we add the hundred's column 1(the carry), 1 and 4 getting 6 as a sum. Put the 6 under the hundred's column for a final sum of 635. Six hundred and thirty five.
Add the one's column of 2 and 8 resulting in 10. Place the one's part of 10 under the one's column. Carry the ten's part of 10 over to be added in the ten's column. Add the ten's column 1, 3 and 4 for a tally of 8. Place the one's part of 8 under the ten's column. There isn't any ten's part of 8 to carry. Add the hundred's column of 4 and 8 resulting in 12. Put the one's part of 12 under the hundred's column. Carry the ten's part of 12 over to the thousand's column. There are no numbers in the thousand's column so bring the 1 down and place it in the thousand's place of the sum. The final answer is 1280. One thousand two hundred and eighty.
Copyright 2008 Robert Lee Thomas
18218 Fewins Rd. Interlochen, MI 49643
Raombyert Distributor
My e-mail robert@homemadesoftware.com
http://www.homemadesoftware.com/